# Ordering is Preserved on Integers by Addition

## Theorem

The usual ordering on the integers is preserved by the operation of addition:

$\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$

## Proof

Recall that Integers form Ordered Integral Domain.

$\forall x, y, z \in \Z: x \le y \implies \paren {x + z} \le \paren {y + z}$
$\forall x, y, z \in \Z: x \le y \implies \paren {z + x} \le \paren {z + y}$

So:

 $\displaystyle a$ $\le$ $\displaystyle b$ $\displaystyle \leadsto \ \$ $\displaystyle a + c$ $\le$ $\displaystyle b + c$ Relation Induced by Strict Positivity Property is Compatible with Addition

 $\displaystyle c$ $\le$ $\displaystyle d$ $\displaystyle \leadsto \ \$ $\displaystyle b + c$ $\le$ $\displaystyle b + d$ Relation Induced by Strict Positivity Property is Compatible with Addition

Finally:

 $\displaystyle a + c$ $\le$ $\displaystyle b + c$ $\displaystyle b + c$ $\le$ $\displaystyle b + d$ $\displaystyle \leadsto \ \$ $\displaystyle a + c$ $\le$ $\displaystyle b + d$ Definition of Ordering

$\blacksquare$